Coprimality Density and the Proper-Solution Deficit in the Generalized Fermat Family {2, 3,m}

We compute the coprimality density for the generalized Fermat family {2,3,m}, addressing the open problem of the companion census [Ross] (seven coprime solutions observed against a raw density-model expectation of 33.04). The coprimality correction is resolved exactly, a rank-zero component of the deficit is proved, and the positive-rank remainder is measured and framed as a hypothesis. The anchor-divisor properness correction exists in closed form and is per-anchor, not universal. Properness of a solution of x³ ± a^m = y² is exactly the condition gcd(x,a) = 1, with local density ℓ(a) = ∏_{p | a} ℓ_p, ℓ_p = 1 - 1/p for odd p and ℓ_2 = 1/2, reducing the raw expectation to 19.25. The local densities at 2 and 3 are likewise closed-form, and depend only on the anchor's residue modulo 8 and 9. Finite-cutoff local diagnostics — exact 2- and 3-adic densities and good-prime point-count products through z ≤ 3000 — increase rather than decrease the weighted total (to 24.5–27.5 over the tested cutoffs), so every tested truncation of this local diagnostic moves the expected total upward rather than toward the observation. No limit of the prime product is claimed; the rigid rank-zero example exhibits substantial decay. A layer reduction of the anchor ensemble by sixth-power-free class proves part of the deficit outright. The curves collapse to 670 distinct classes whose Mordell–Weil ranks we determine unconditionally by 2-descent (264 of rank 0, 337 of rank 1, 66 of rank 2, 3 of rank 3). Rank-zero classes carry 12.1% of the diagnostic mass and provably contribute no proper solutions, including the 28 anchors with 6 | m, whose curves are globally the rank-zero curves y² = X³ ± 1. The local corrections carry real locational information: ranked by the raw model alone, two of the seven solution-bearing anchors fall in the top decile and, ranked by the full diagnostic, six do. The dominant remaining shortfall sits on positive-rank classes, consistent with the height-gap phenomenology of quantitative Siegel bounds; we state this as a hypothesis, not a conclusion.